I have several bantams that won't stop going broody. After I kept removing eggs, one of them snuck under the shed and raised 6 babies. Then another 5 after that. I'm ready to make broody stew, lol. So I'd recommend bantams.
Its a possibility. Leghorns do tend to be a more aggressive breed. You don't necessarily need to go the bantam route to get a broody, though. Breeds like standard Cochins, Buff Orpingtons, Austrolorps, Games and many others are notorious for being setters. And, are much less likely to be pushed around by your Leghorns. Or, you could just buy an incubator, then you could hatch eggs whenever you wanted, without having to wait on a hen to sit on them
It's getting late in the year for a hen to go broody. It does happen occasionally through the winter, but don't count on it and when you are wanting a hen to go broody, it's like a watched pot, it never boils, so don't expect to get one of the suggested breeds and for it to go broody and raise chicks any time soon.
Firstly it will need to settle in and start laying, which usually takes 2-4 weeks, then it will lay for several weeks, to produce a clutch of eggs, before it gets to the point of wanting to set. By that time we will be into winter and that will probably suppress the broody instinct, so you may be looking at Spring/summer next year before your mother hen is actually in broody mode. You might also find the one you buy, doesn't go broody at all. Just because a breed of hen is prone to it, doesn't mean an individual will go broody.
Apologies if you are already aware of these facts, it just sounded like you were after a quick fix and much as I love hatching chicks with a broody hen, they do it in their own good time. If you decide to go for an incubator and raise some now, be aware that you will need to keep the chicks warm through the winter.